If V={(x,y)|x,y ∈ R} with acts (x1,y1)@(x2,y2)=\((\sqrt[3]{(x1^3+x2^3)},\sqrt[3]{(y1^3+y2^3)})\) and a$(x1,y1)=(ax1,ay1) , a ∈ R.

Prove V with this acts is **NOT** vector space.Thank you!

Dimitristhym Nov 12, 2018

#1**+1 **

are you sure it's not?

it's addition has commutativity and associativity

(0,0) is the clear additive identity and (-x,-y) is the additive inverse of (x,y)

the scalar properties are identical to the Euclidean vector space

I'm not seeing why this isn't a vector space.

remember the cube and cube root form a bijection

Rom Nov 12, 2018

#2**0 **

This is not a vector space. Exactly one of the axioms of a vector space is not satisfied.

Guest Nov 12, 2018

#3**+1 **

Rom yes im sure the exercise say it's **not.**For this reason i can't solve the exercise

Dimitristhym
Nov 12, 2018

#4**+1 **

Guest "Exactly one of the axioms of a vector space is not satisfied"

Witch is this axiom?

Dimitristhym
Nov 12, 2018

#5**+3 **

Ok. Distribution of scalar mutliplication with respect to field addition doesn't hold.

we should have

\((a+b)\vec{v} = a \vec{v} \oplus b \vec{v}\)

let's take a look

\((a+b)\vec{v} =\{(a+b)v_x,(a+b)v_y\}\\ a\vec{v} \oplus b \vec{v} =\left \{\sqrt[3]{a^3 v_x^3+b^3 v_x^3}, \sqrt[3]{a^3 v_y^3+b^3 v_y^3 }\right \}=\\ \left\{\sqrt[3]{a^3+b^3}v_x,~\sqrt[3]{a^3+b^3}v_y \right\}\\ \sqrt[3]{a^3+b^3} \neq a+b\)

Rom
Nov 13, 2018